2016/04/06 by Cianchi, Andrea, Ferone, Vincenzo, Nitsch, Carlo +1
#26B30 #46E35 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1604.01524
Extremal functions are exhibited in Poincaré trace inequalities for functions of bounded variation in the unit ball \mathbb Bn of the n-dimensional Euclidean space \mathbb Rn. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of \mathbb Bn, instead of just on ∂ \mathbb Bn, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the latter constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain Ω⊂ \mathbb Rn, in terms of an isoperimetric inequality for subsets of Ω.